Answer
The graph is shown below:
Work Step by Step
Let us consider the inequality $ y\le -\frac{1}{2}x+2$; substitute the equal symbol in place of the inequality and rewrite the equation as given below:
$ y=-\frac{1}{2}x+2$
To find the value of the x-intercept, put y = 0 as given below:
$\begin{align}
& 0=-\frac{1}{2}x+2 \\
& \frac{1}{2}x=2 \\
& x=4
\end{align}$
To find the value of the y-intercept, substitute x = 0 as given below:
$\begin{align}
& y=\left( -\frac{1}{2}\times 0 \right)+2 \\
& =2
\end{align}$
Then, take the origin $\left( 0,0 \right)$ as a test point and check the region in the graph to shade:
$\begin{align}
& y\le -\frac{1}{2}x+2 \\
& y+\frac{1}{2}x\le 2 \\
& 0+\left( \frac{1}{2}\times 0 \right)\le 2 \\
& 0\le 2
\end{align}$
So, the above expression is correct, which means the shaded region will contain the test point; that is, the region towards the origin will be shaded. Therefore, the line passes through points $\left( 4,0 \right)$ and $\left( 0,2 \right)$.
Plot the graph using the intercepts as given below:
Thus, the graph of the provided inequality is plotted and the line passes through points $\left( 4,0 \right)$ and $\left( 0,2 \right)$. Also, we see that the shaded region contains the origin.
